Hermite's Identity

Relevant For...

Hermite's identity is an identity that computes the value of a summation involving floor functions. It is named after Charles Hermite, a French mathematician who did research during the nineteenth century. It is useful in solving equations, proving inequalities, computing sums etc. involving floor functions.

Notice that there is no restriction on the sign of \(x,\) i.e. it can be positive or negative.

Proof

Before presenting the formal proof, one idea of the proof has to be noted. The key fact here is that \(\lfloor x \rfloor,\left \lfloor x + \frac{1}{n} \right \rfloor,\ldots, \left \lfloor x + \frac{n - 1}{n} \right \rfloor\) can take only two values, namely \(\lfloor x \rfloor\) and \(\lfloor x \rfloor+1\). Showing this is not too hard. Note that

Find \(\lfloor 100r \rfloor\).

The given sum has exactly \(91-19+1=73\) terms, each of which equals \(\lfloor r \rfloor\) or \(\lfloor r \rfloor+1\). If all of the terms equal \(7\), then the total sum is \(7\cdot 73=521,\) which is less than \(546\). If all of the terms equal \(8\), then the total sum is \(8\cdot 73=581\), which is greater than \(546\). It follows that some of the terms take the value \(7\) while the others take \(8\). This means that \(\lfloor r \rfloor=7\). Suppose the first \(i\) terms take the value \(7\) whie the other \(73-i\) terms take the value \(8\). Then, \(7i+8(73-i)=546\). Solving this, we get \(i=38\). This means that \(\left\lfloor r + \frac{56}{100} \right\rfloor=7\) and \(\left\lfloor r + \frac{57}{100} \right\rfloor=8\). Therefore,

It must be admitted that there are easier ways to prove this inequality.

Use the fact that \(\lfloor r \rfloor+\left\lfloor r+\frac12\rfloor=\lfloor2r \right\rfloor\) and the given problem reduces to proving the inequality \(\left\lfloor x+\frac12\right\rfloor+\left\lfloor y+\frac12\right\rfloor\ge \lfloor x+y\rfloor\). This can be proven by substituting \(x=\lfloor x \rfloor +\{x\}\) and \(y=\lfloor y \rfloor +\{y\}\) and considering what the fractional parts of \(x\) and \(y\) can be. This part is left to the reader.